Explain futures pricing and linear regression basics
Company: Morgan Stanley
Role: Data Scientist
Category: Machine Learning
Difficulty: hard
Interview Round: Technical Screen
You are interviewing for a quantitative/strats role. The interviewer asks a series of theoretical questions about derivatives pricing and linear regression.
1. **Futures/Forwards Pricing**
Assume an idealized (frictionless) market with:
- An underlying asset with current spot price S0
- A continuously compounded risk‑free interest rate r
- Contract maturity T (in years)
- No storage costs, no income/dividends from the asset, and no transaction costs or arbitrage opportunities
Answer the following:
(a) Using a no‑arbitrage argument, derive the theoretical fair forward/futures price F0 and show that:
F0 = S0 * exp(r * T).
(b) Explain the *intuition* behind this formula: why should the futures price have this relationship to the spot price and the risk‑free interest rate?
(c) Briefly explain, at a high level, how similar replication / no‑arbitrage ideas are used to price simple European options on the same underlying. You do **not** need to derive a full closed‑form formula like Black–Scholes, but you should explain the overall approach (for example, via constructing a replicating portfolio or using risk‑neutral valuation).
2. **Linear Regression and OLS**
Consider the standard linear regression model:
y = X * beta + epsilon
where:
- y is an n-dimensional response vector
- X is an n x p design matrix with full column rank
- beta is a p-dimensional vector of unknown parameters
- epsilon is an n-dimensional error term
Answer the following:
(a) State the classical assumptions underlying ordinary least squares (OLS) linear regression (for example, assumptions about linearity, independence, error mean, homoscedasticity, absence of multicollinearity, etc.).
(b) Derive the closed‑form expression for the OLS estimator beta_hat by minimizing the sum of squared residuals:
minimize over beta: ||y − X * beta||^2.
Show the main steps and give the final matrix formula for beta_hat.
(c) In simple linear regression with one predictor (y_i = alpha + beta * x_i + epsilon_i), write the closed‑form expression for the slope coefficient beta_hat in terms of sample covariances and variances (for example, using Cov(x, y) and Var(x)).
(d) Briefly state what additional assumptions are needed for:
- beta_hat to be the Best Linear Unbiased Estimator (BLUE), and
- standard inference procedures (such as t‑tests and confidence intervals) about beta to be valid.
Quick Answer: This question evaluates understanding of no‑arbitrage derivatives pricing principles and core linear regression (OLS) estimation, assessing intuition about forward/futures valuation and replication/risk‑neutral ideas alongside formulation and assumptions of OLS estimators.